Question

Find the equation of the exponential function represented by the table below:
xx yy
00 44
11 1616
22 6464
33 256256

Answer 1

To find the equation of the exponential function represented by the given table, we’ll look for a function of the form:

\[
y = ab^x
\]

where \( a \) is the initial value (i.e., the value when \( x = 0 \)), and \( b \) is the growth factor. The table data you've provided is as follows:

- For \( x = 0 \), \( y = 4 \)
- For \( x = 1 \), \( y = 16 \)
- For \( x = 2 \), \( y = 64 \)
- For \( x = 3 \), \( y = 256 \)

From the first entry, we can immediately see that:

\[
a = 4
\]

This gives us the equation:

\[
y = 4b^x
\]

Next, we will use the other points to determine the value of \( b \).

Using the point \( (1, 16) \):

\[
16 = 4b^1
\]

Solving for \( b \):

\[
b = \frac{16}{4} = 4
\]

So now we have:

\[
y = 4 \cdot 4^x
\]

To confirm our function, we can check the other points:

For \( x = 2 \):

\[
y = 4 \cdot 4^2 = 4 \cdot 16 = 64
\]

For \( x = 3 \):

\[
y = 4 \cdot 4^3 = 4 \cdot 64 = 256
\]

Both values match the provided table.

Thus, the equation of the exponential function represented by the table is:

\[
y = 4 \cdot 4^x
\]